Table 1.

The four different cloak design.

	Coordinate transformation	Impedance function k	k value at boundary r = R2	Parameters of the transformation media
Case 1	$r^{\prime }=f(r)=\frac{R_2}{R_2-R_1}r-\frac{R_1{R}_2}{R_2-R_1}$	1	1	$\frac{1}{{\rho }_r}=\frac{r-R_1}{r}\frac{1}{{\rho }_0},$ $\frac{1}{{\rho }_{\theta }}=\frac{r}{r-R_1}\frac{1}{{\rho }_0},$ $\frac{1}{\lambda }={\left(\frac{R_2}{R_2-R_1}\right)}^2\frac{r-R_1}{r}\frac{1}{{\lambda }_0}$
Case 2	$r^{\prime }=f(r)=\frac{R_2}{R_2-R_1}r-\frac{R_1{R}_2}{R_2-R_1}$	$k={(\frac{R_2}{R_2-R_1})}^2\frac{r-R_1}{r}$	$\frac{R_2}{R_2-R_1}\ne 1$	$\frac{1}{{\rho }_r}={\left(\frac{R_2-R_1}{R_2}\right)}^2\frac{1}{{\rho }_0},$ $\frac{1}{{\rho }_{\theta }}={\left(\frac{R_2-R_1}{R_2}\right)}^2{\left(\frac{r}{r-R_1}\right)}^2\frac{1}{{\rho }_0},$ $\frac{1}{\lambda }=\frac{1}{{\lambda }_0}$
Case 3	$r^{\prime }=f(r)=\frac{R_2}{R_2-R_1}r-\frac{R_1{R}_2}{R_2-R_1}$	$k=\frac{R_2}{R_2-R_1}\frac{r-R_1}{r}$	1	$\frac{1}{{\rho }_r}=\frac{R_2-R_1}{R_2}\frac{1}{{\rho }_0},$ $\frac{1}{{\rho }_{\theta }}=\frac{R_2-R_1}{R_2}{\left(\frac{r}{r-R_1}\right)}^2\frac{1}{{\rho }_0},$ $\frac{1}{\lambda }=\frac{R_2}{R_2-R_1}\frac{1}{{\lambda }_0}.$
Case 4	$r^{\prime }=f(r)=A{r}^2+Br+c$	$k=\frac{A{r}^2+Br+C}{r}(2Ar+B)$	1	$\frac{1}{{\rho }_r}=\frac{1}{{(2Ar+B)}^2}\frac{1}{{\rho }_0},$ $\frac{1}{{\rho }_{\theta }}=\frac{r^2}{{(A{r}^2+Br+C)}^2}\frac{1}{{\rho }_0},$ $\frac{1}{\lambda }=\frac{1}{{\lambda }_0}$